Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

$ \begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}} {{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }} {{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered} $ \begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}} {{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }} {{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered}
Keywords:	Estimates of all homogeneous expansions  Quasi-convex mapping  Quasi-convex mapping of type A  Quasi-convex mapping of type B
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